CS 201 Data Structures and Discrete Mathematics I

CS 201: Data Structures and Discrete Mathematics I Basic Set Theory 12/4/2020 CS 201 1

Sets • A set is a collection of distinct objects. • For example (let A denote a set): A = {apple, pear, grape} A = {1, 2, 3, 4, 5} A = {1, b, c, d, e, f} A = {(1, 2), (3, 4), (9, 10)} A = {<1, 2, 3>, <3, 4, 5>, <6, 7, 8>} A = a collection of anything that is meaningful. 12/4/2020 CS 201 2

Members and Equality of Sets • The objects that make up a set are called members or elements of the set. • Two sets are equal iff they have the same members. – That is, a set is completely determined by its members. – Order and repetition do not matter in a set. 12/4/2020 CS 201 3

Set notations • The notation of {. . . } describes a set. Each member or element is separated by a comma. – E. g. , S = {apple, pear, grape} – S is a set – The members of S are: apple, pear, grape • Order and repetition do not matter in a set. • The following expressions are equivalent: – {1, 3, 9} – {9, 1, 3} – {1, 1, 3, 3, 9} 12/4/2020 CS 201 4

The membership symbol and the empty set • The fact that x is a member of a set S can be expressed as – x S – Reads: • x is in S, or • x is a member of S, or • X belongs to S • An example, S = {1, 2, 3}, 1 S, 2 S, 3 S • The negation of is written as (is not in). • The empty set is a set without any element – Denoted by {} or – For any object x, x 12/4/2020 CS 201 5

Defining a Set by membership properties • Notation – S = {x T | P(x)} (or S = {x | x T and P(x)}) – The members of S are members of an already known set T that satisfy property P. • An example: – Let Z be the set of integers – Let Z+ be the set of positive integers. – Z+ = {x Z | x > 0} 12/4/2020 CS 201 6

Sets of numbers • Z = The set of all integers Z = {…, -2, -1, 0, 1, 2, …} • Z+ = The set of positive integers Z+ = {1, 2, 3…} = {x | x Z and x > 0} = {x Z | x > 0} • Z- = The set of negative integers Z- = {…, -3, -2, -1} = {-1, -2, -3…} = {x Z | x < 0} • R = The set of all real numbers • Q = the set of all rational numbers Q = {x R | x = p/q and p, q Z and q 0} • We can use “; ” to replace “and” 12/4/2020 CS 201 7

Subsets • A is a subset ( ) of B, or B is a superset of A iff every member of A is a member of B. – A B iff forall x if x A, then x B • An example: – (-2, 0, 6} {-3, -2, -1, 0, 1, 3, 6} • Negation: A is not a subset of B or B is not a superset of A iff there is a member of A that is not a member of B – A B iff there exist x, x A, x B 12/4/2020 CS 201 8

Obvious subsets –S S – S • By contradiction: • if S then there exist x, x and x S. 12/4/2020 CS 201 9

Proper subsets • A is a proper subset ( ) of B, or B is a proper superset of A iff A is a subset of B and A is not equal to B. – A B iff A B and A B • Examples: – {1, 2, 3} {1, 2, 3, 4, 5} – Z+ Z Q R – If S then S 12/4/2020 CS 201 10

Power sets • The set of all subsets of a set is called the power set of the set • The power set of S is denoted by P(S). • Example: – – P( ) ={ } P({1, 2}) = { , {1}, {2}, {1, 2}} P(S) = { , …, S} What is P({1, 2, 3})? • How many elements does the power set of S have? Assume S has n elements. 12/4/2020 CS 201 11

and are different • Examples: 1 {1} is true 1 {1} is false {1} is true • Which of the following statement is true? S P(S) 12/4/2020 CS 201 12

Mutual inclusion and set equality • Sets A and B have the same members iff they mutually include – A B and B A • That is, A = B iff A B and B A • Mutual inclusion is very useful for proving the equality of sets • To prove an equality, we prove two subset relationships. 12/4/2020 CS 201 13

An example: equality of sets • • Recall that Z = the set of (all) integers Let A = {x Z | x = 2 m for m Z } Let B = {x Z | x = 2 n-2 for n Z } To show A B, note that 2 m = 2(m+1) – 2 = 2 n-2 • To show that B A, note that 2 n-2=2(n-1) = 2 m • That is, A = B. (A, B both denote the set of all even integers. 12/4/2020 CS 201 14

Universal sets • Depending on the context of discussion – Define a set of U such that all sets of interest are subsets of U. – The set U is known as a universal set • Examples: – When dealing with integers, U may be Z. – When dealing with plane geometry, U may be the set of points in the plane 12/4/2020 CS 201 15

Venn diagram • Venn diagram is used to visualize relationships of some sets. • Each subset (of U, the rectangle) is represented by a circle inside the rectangle. 12/4/2020 CS 201 16

Set operations • Let A, B be subsets of some universal set U. • The following set operations create new sets from A and B. • Union: A B = {x U | x A or x B} • Intersection: A B = {x U | x A and x B} • Difference: A B = A B= {x U | x A and x B} • Complement A = U A = {x U | x A} 12/4/2020 CS 201 17

Set union • An example {1, 2, 3} {1, 2, 4, 5} = {1, 2, 3, 4, 5} The venn diagram 3 12/4/2020 1 5 2 4 CS 201 18

Set intersection • An example {1, 2, 3} {1, 2, 4, 5} = {1, 2} The venn diagram 3 12/4/2020 1 5 2 4 CS 201 19

Set difference • An example {1, 2, 3} - {1, 2, 4, 5} = {3} The venn diagram 3 12/4/2020 1 5 2 4 CS 201 20

Set complement • The venn diagram 12/4/2020 CS 201 21

Some questions • Let A B. – What is A – B? – What is B – A? – If A, B C, what can you say about A B and C? – If C A, B, what can you say about C and A B? 12/4/2020 CS 201 22

Basic set identities (equalities) • Commutative laws A B=B A • Associative laws (A B) C = A (B C) • Distributive laws A (B C) = (A B) (A C) 12/4/2020 CS 201 23

Basic set identities (cont …) • Identity laws A=A =A A U=U A=A • Double complement law (A’)’ = A • Idempotent laws A A=A • De Morgan’s laws (A B)’ = A’ B’ 12/4/2020 CS 201 24

Basic set identities (cont …) • Absorption laws A (A B) = A • Complement law (U)’ = U • Set difference law A – B = A B’ • Universal bound law A U=U A = 12/4/2020 CS 201 25

Proof methods • There are many ways to prove set identities • The methods include – Applying existing identities – Using mutual inclusion • Prove (A B) C = A (B C) using mutual inclusion – First show: (A B) C A (B C) – Let x (A B) and x C • (x A and x B) and x C • x A and x (B C) • x A (B C) – Then show that A (B C) (A B) C 12/4/2020 CS 201 26

More proof examples • Let B = {x | x is a multiple of 4} A = {x | x is a multiple of 8} Then we have A B • Proof: let x A. We must show that x is a multiple of 4. We can write x = 8 m for some integer m. We have – x = 8 m = 2*4 m = 4 k, where k = 2 m, – so k is a integer. – Thus, x is a multiple of 4, and • therefore x B 12/4/2020 CS 201 27

More proof examples • Prove {x | x Z and x 0 and x 2 < 15} = {x | x Z and x 0 and 2 x < 7} Proof: Let A = {x | x Z and x 0 and x 2 < 15} B = {x | x Z and x 0 and 2 x < 7} Let x A. x can only be 0, 1, 2, 3 2 x for 0, 1, 2, 3 are all less then 7. Thus, A B. Likewise, we can also show that B A 12/4/2020 CS 201 28

Algebraic proof examples • Prove: [A (B C)] ([A’ (B C)] (B C)’) = Proof: [A (B C)] ([A’ (B C)] (B C)’) = ([A (B C)] [A’ (B C)]) (B C)’ (associative) = ([(B C) A] [(B C) A’]) (B C)’ (commutative) = [(B C) (A A’)] (B C)’ (distributive) = [(B C) ] (B C)’ (complement) = (B C)’ (Identity) = (identity) 12/4/2020 CS 201 29

Algebraic proof examples • Prove: (A B) – C = (A – C) (B – C) • Proof: (A B) – C = (A B) C’ (difference) = C’ (A B) (commutative) = (C’ A) (C’ B) (distributive) = (A C’) (B C’) (commutative) = (A – C) (B – C) (difference) 12/4/2020 CS 201 30

Disproving an alleged Set property • Is the following true? (A – B) (B – C) = A – C Solution: Draw a Venn diagram and construct some sets to confirm the answer Counterexample: A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, and C = {4, 5, 6, 7} A – B = {1, 4}, B – C = {2, 3}, A – C = {1, 2} (A – B) (B – C) = {1, 2, 3, 4} 12/4/2020 CS 201 31

Pigeonhole principle • If more than k pigeons fly into k pigeonholes, then at least one hole will have more than one pigeon. • Pigeonhole principle: if more than k items are placed into k bins, then at least one bin contains more than one item. • Simple, and obvious!! • To apply it, may not be easy sometimes. Need to be clever in identifying pigeons and pigeonholes. 12/4/2020 CS 201 32

Example • How many people must be in a room to guarantee that two people have last names that begin with the same initial? 27 since we have 26 letters • How many times must a single die be rolled in order to guarantee getting the same value twice? 7 12/4/2020 CS 201 33

Another example • Prove that if four numbers are chosen from the set {1, 2, 3, 4, 5, 6}, at least one pair must add up to 7. Proof: There are 3 pairs of numbers from the set that add up to 7, i. e. , (1, 6), (2, 5), (3, 4) Apply pigeonhole principle: bins are the pairs, and the numbers are the items. 12/4/2020 CS 201 34

Infinite sets • In a finite set, we can always designate one element as the first member, s 1, another element as the second member, s 2 and so on. If there are k elements in the set we can list them as – s 1, s 2, …, sk • A set that is not finite is called a infinite set. • If a set is infinite, we may still be able to select a first element s 1, a second element s 2 and so on: – s 1, s 2, … – Such an infinite set is said to be denumerable. • Both finite and denumerable sets are countable. • Countable does not mean we can give a total number, but means that we can say, “here is the first one” and “here is the second one” and so on. 12/4/2020 CS 201 35

Countable sets: examples • The set of positive integer numbers are countable. • To prove it, we need to give a counting scheme, in this case, 1, 2, 3, 4, …. • The set of positive rational numbers are countable 1/1 1/2 1/3 1/4 … 2/1 2/2 2/3 2/4… 3/1 3/2 3/3 3/4 … 12/4/2020 CS 201 36

Uncountable sets • There also some sets that are uncountable. – The set is so large, and there is no way to count out the elements. • One example: The set of real numbers between 0 and 1 is uncountable. • A computer can only manage finite sets. 12/4/2020 CS 201 37

Summary • Sets are extremely important for Computer Science. • A set is simply an unordered list of objects. • Set operations: union, intersection, difference. • To prove set equalities – Applying existing identities – Using mutual inclusion • Pigeonhole principle 12/4/2020 CS 201 38